When you go out looking for lost puppies and return them to the Rescue shelter the first thing to happen is you get a random mini-game. The mini-game can be vaccination, remove fleas or clean the teeth. Now you shouldn't worry about this achievement or the remove 15 fleas one before they are the last achievements you need.
You will get some of these (if you get lucky you might get them all) done while going for the other achievements. The mini-game is very easy. You first need to take the vaccine into the syringe by pressing
, then you need to press
when the red target lines up with the blue ring and finally you shoot the vaccine into the puppy's mouth with
My tip for this: empty all your kennels (you should have 6 plus Barbie's kennel) and go look for new puppies. When you return to the Shelter, IF you don't get the Vaccine mini-game or the Flea mini-game, just dashboard by pressing
. Then launch the game again and load up your save game. Then go look for another puppy and hope to get a mini-game you still need for the achievements. This way you won't be filling your kennel with the puppies that won't give you the needed mini-game. Because if you fill up all your kennels you need to complete all the other grooming and training mini-games again to get those puppies adopted and to free more space.